Twisted 𝐾-theory and Poincaré duality

Tu, Jean-Louis

doi:10.1090/S0002-9947-08-04706-5

Twisted $K$-theory and Poincaré duality
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by Jean-Louis Tu PDF

Trans. Amer. Math. Soc. 361 (2009), 1269-1278 Request permission

Abstract:

Using methods of $KK$-theory, we generalize Poincaré $K$-duality to the framework of twisted $K$-theory.

References

Michael Atiyah and Graeme Segal, Twisted $K$-theory, Ukr. Mat. Visn. 1 (2004), no. 3, 287–330; English transl., Ukr. Math. Bull. 1 (2004), no. 3, 291–334. MR 2172633
Jacek Brodzki, Varghese Mathai, Jonathan Rosenberg, and Richard J. Szabo, D-branes, RR-fields and duality on noncommutative manifolds, Comm. Math. Phys. 277 (2008), no. 3, 643–706. MR 2365448, DOI 10.1007/s00220-007-0396-y
U. Bunke and I. Schröder, Twisted $K$-theory and TQFT, Mathematisches Institut, Georg-August-Universität Göttingen: Seminars Winter Term 2004/2005, Universitätsdrucke Göttingen, Göttingen, 2005, pp. 33–80. MR 2206878
A. Connes and G. Skandalis, The longitudinal index theorem for foliations, Publ. Res. Inst. Math. Sci. 20 (1984), no. 6, 1139–1183. MR 775126, DOI 10.2977/prims/1195180375
Carey, A.L. and Wang, B.L., Thom isomorphism and Push-forward map in twisted $K$-theory. Preprint. arXiv:math/0512139
P. Donovan and M. Karoubi, Graded Brauer groups and $K$-theory with local coefficients, Inst. Hautes Études Sci. Publ. Math. 38 (1970), 5–25. MR 282363
Echterhoff, S., Emerson, H. and Kim, H.J. KK-theoretic duality for proper twisted actions. arXiv:math/0610044 Preprint.
Daniel S. Freed, Michael J. Hopkins, and Constantin Teleman, Twisted equivariant $K$-theory with complex coefficients, J. Topol. 1 (2008), no. 1, 16–44. MR 2365650, DOI 10.1112/jtopol/jtm001
Freed, D., Hopkins, M. and Teleman, C. Loop groups and twisted $K$-theory II. arXiv:math.AT/0206257.
Michel Hilsum and Georges Skandalis, Morphismes $K$-orientés d’espaces de feuilles et fonctorialité en théorie de Kasparov (d’après une conjecture d’A. Connes), Ann. Sci. École Norm. Sup. (4) 20 (1987), no. 3, 325–390 (French, with English summary). MR 925720
Max Karoubi, Algèbres de Clifford et $K$-théorie, Ann. Sci. École Norm. Sup. (4) 1 (1968), 161–270 (French). MR 238927
Karoubi, M. Twisted K-theory, old and new. To appear in the Proceedings of the Valladolid conference (2007), European Mathematical Society.
G. G. Kasparov, The operator $K$-functor and extensions of $C^{\ast }$-algebras, Izv. Akad. Nauk SSSR Ser. Mat. 44 (1980), no. 3, 571–636, 719 (Russian). MR 582160
G. G. Kasparov, Equivariant $KK$-theory and the Novikov conjecture, Invent. Math. 91 (1988), no. 1, 147–201. MR 918241, DOI 10.1007/BF01404917
Pierre-Yves Le Gall, Théorie de Kasparov équivariante et groupoïdes. I, $K$-Theory 16 (1999), no. 4, 361–390 (French, with English and French summaries). MR 1686846, DOI 10.1023/A:1007707525423
Ellen Maycock Parker, The Brauer group of graded continuous trace $C^*$-algebras, Trans. Amer. Math. Soc. 308 (1988), no. 1, 115–132. MR 946434, DOI 10.1090/S0002-9947-1988-0946434-7
Jean-Louis Tu, Ping Xu, and Camille Laurent-Gengoux, Twisted $K$-theory of differentiable stacks, Ann. Sci. École Norm. Sup. (4) 37 (2004), no. 6, 841–910 (English, with English and French summaries). MR 2119241, DOI 10.1016/j.ansens.2004.10.002